SOME RESULTS IN THE REPRESENTATION THEORY OF STRONGLY GRADED VERTEX ALGEBRAS

Abstract

OF THE DISSERTATION Some results in the representation theory of strongly graded vertex algebras By JINWEI YANG Dissertation Director: Yi-Zhi Huang and James Lepowsky In the first part of this thesis, we study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We consider a tensor product of strongly graded vertex algebras and its tensor product strongly graded modules. We prove that a tensor product of strongly graded irreducible modules for a tensor product of strongly graded vertex algebras is irreducible, and that such irreducible modules, up to equivalence, exhaust certain naturally defined strongly graded irreducible modules for a tensor product of strongly graded vertex algebras. We also prove that certain naturally defined strongly graded modules for the tensor product strongly graded vertex algebra are completely reducible if and only if every strongly graded module for each of the tensor product factors is completely reducible. These results generalize the corresponding known results for vertex operator algebras and their modules. In the second part, we derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under